Most people today use the terms Private cloud and Internal cloud interchangeably. I see them differently.
I consider an "internal cloud" to a cloud computing setup on on-premise data-center of an Organization. In this case the organization owns the infrastructure and uses the cloud computing platform for the optimal usage of its infrastructure in its data-center.
The motivation behind an internal cloud is data-center consolidation using virtualization and dynamic provisioning technologies to allow
a) optimal usage of infrastructure resources
b) near real-time response to varying need of application capacity (elastic scalability).
An internal cloud is almost always in form of IaaS.
A Private Cloud does not necessarily refer to an internal data-center consolidation. It may refer to set of services provided exclusively to a customer. In case of private cloud a vendor will provide a cloud computing platform to a customer that has been tailored to custom needs of the customer. This customer has exclusive access to the cloud. A private cloud is generally not multi-tenants in the sense that the resources will not be shared with other customers.
A private cloud may also refer to cloud computing platform that has been tailored to an exclusive community of users (such as law firms, universities etc.). In this type of setup the cloud can be multi-tenant.
Private clouds generally offer higher level of compliance and governance than a public cloud.
A private cloud can in the form of IaaS, PaaS or even SaaS.
Private cloud and Internal cloud share some of the characteristics. The resources are not shared among multiple organizations. Provide higher degree of control over data governance and regulation compliance.